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The |$2$| -layer drawing model is a well-established paradigm to visualize bipartite graphs where vertices of the two parts lie on two horizontal lines and edges lie between these lines. Several beyond-planar graph classes have been studied under this model. Surprisingly, however, the fundamental class of |$k$| -planar graphs has been considered only for |$k=1$| in this context. We provide several contributions that address this gap in the literature. First, we show tight density bounds for the classes of |$2$| -layer |$k$| -planar graphs with |$k\in \{2,3,4,5\}$|⁠. Based on these results, we provide a Crossing Lemma for |$2$| -layer |$k$| -planar graphs, which then implies a general density bound for |$2$| -layer |$k$| -planar graphs. We prove this bound to be almost optimal with a corresponding lower bound construction. Finally, we study relationships between |$k$| -planarity and |$h$| -quasiplanarity in the |$2$| -layer model and show that |$2$| -layer |$k$| -planar graphs have pathwidth at most |$k+1$| while there are also |$2$| -layer |$k$| -planar graphs with pathwidth at least |$(k+3)/2$|⁠. [ABSTRACT FROM AUTHOR]

The circumference of a graph G with at least one cycle is the length of a longest cycle in G. A classic result of Birmel\'e [J. Graph Theory, 43 (2003), pp. 24--25] states that the treewidth of G is at most its circumference minus 1. In case G is 2-connected, this upper bound also holds for the pathwidth of G; in fact, even the treedepth of G is upper bounded by its circumference (Bria\'nski et al. [Treedepth vs circumference, Combinatorica, 43 (2023), pp. 659--664]). In this paper, we study whether similar bounds hold when replacing the circumference of G by its cocircumference, defined as the largest size of a bond in G, an inclusionwise minimal set of edges F such that G F has more components than G. In matroidal terms, the cocircumference of G is the circumference of the bond matroid of G. Our first result is the following "dual" version of Birmel'e's theorem: The treewidth of a graph G is at most its cocircumference. Our second and main result is an upper bound of 3k 2 on the pathwidth of a 2-connected graph G with cocircumference k. Contrary to circumference, no such bound holds for the treedepth of G. Our two upper bounds are best possible up to a constant factor. [ABSTRACT FROM AUTHOR]

Given two n-vertex graphs G1 and G2 of bounded treewidth, is there an n-vertex graph G of bounded treewidth having subgraphs isomorphic to G1 and G2? Our main result is a negative answer to this question, in a strong sense: we show that the answer is no even if G1 is a binary tree and G2 is a ternary tree. We also provide an extensive study of cases where such "gluing"" is possible. In particular, we prove that if G1 has treewidth k and G2 has pathwidth \l, then there is an n-vertex graph of treewidth at most k + 3\l + 1 containing both G1 and G2 as subgraphs. [ABSTRACT FROM AUTHOR]